Complex Numbers (Book)

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Complex Numbers

Complex Numbers

A complex number is expressed in the standard form a + bi, where a and b are real numbers and i is defined by i^2 = -1 (that is, i is the square root of -1). For example, 3 + 2i is a complex number.

The bi term is often referred to as an imaginary number (though this may be misleading, as it is no more "imaginary" than the symbolic abstractions we know as the "real" numbers). Thus, every complex number has a real part, a, and an imaginary part, bi.

Complex numbers are often represented on a graph known as the "complex plane," where the horizontal axis represents the infinity of real numbers, and the vertical axis represents the infinity of imaginary numbers. Thus, each complex number has a unique representation on the complex plane: some closer to real; others, more imaginary. If a = b, the number is equal parts real and imaginary.

Very simple transformations applied to numbers in the complex plane can lead to fractal structures of enormous intricacy and astonishing beauty.

http://www.ddewey.net/mandelbrot/

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Writer_Mike

about 6 hours ago
In response to "Paris, I Guess"

Elisa: Middle Aged Stik

- I thought it very interesting, to say nothing of extraordinarily coincidental, that your video mentions a lock-adorned bridge in Paris. I just finished reading Spymaster by Brad Thor, in which the Russian mastermind takes strolls from Kaliningrad's mainland to Kant Island. The bridge he uses is also festooned with locks placed by couples declaring their undying love for one another.

QPdoll

about 22 hours ago
In response to "Landmarks"

I hope you're enjoying every minute in France! What an opportunity.

2 Like

Wordsmitty ✍️

a day ago
In response to "Landmarks"

Hey! Why are the signs at that Mac's in English if you're in France?

With all that walking, I hope you stayed hydrated.

2 Like

Annette in France

a day ago
In response to "Paris, I Guess"

Bienvenue en France.

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WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

last one today:https://parisjetaime.com/eng/event/olivier-giraud-how-to-become-a-parisian-in-one-ho ur-e047

1 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Could not resist sending the link: https://www.atlasobscura.com/things-to-do/paris-france

1 Like

Elisa: Middle Aged Stik

a day ago
In response to "Paris, I Guess"

Your remarks on Paris looking like many cities reminds me of this video (which I promise is short). It's a sassy look at Paris, and I'm here for it.

https://youtu.be/ByUPFYxy5vw?si=lT57RpTDntAhFdBs

2 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

if you have two hours to spare tomorrow? A beer bike tour in Paris. Check it out, still available. https://parisbeerbike.com/

3 Like

WakeUpAndLive️~Happiness

a day ago
In response to "Paris, I Guess"

Je suis sure, vous avez aimez la ville!!!!

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Aiva Raine

Sep 20, 2024 at 10:13pm
In response to "Get Baked"

FYI- basting them with olive oil before placing them directly on the oven rack- NO tinfoil- will allow the skin to crisp up a little as well and add a nice flavor to the skin. Recommend Idaho baking potatoes as well for best results.

April 19, 2023 at 10:57am

#1048454

Just One Word

Today, another entry for "Journalistic Intentions"

[18+]:

Nylon

"There's a great future in plastics. Think about it. Will you think about it?"
-Mr. McGuire, The Graduate

If Dustin Hoffman's character had actually heeded those words (frankly, I don't remember enough about the movie to know whether he did or not), he might have found himself in trouble. Shortly after the movie was released, "just one word: plastics" started to fall out of favor.

But there was a time when nylon, and other plastics, were heralded as positive agents of change.

As with most innovations of the 20th century, this was largely driven by marketing, with a hint of racism thrown in (nylon was heralded as an alternative to silk, which came from those countries.)

I don't need to rehash the Wikipedia page

for nylon, so for facts (or near-facts) and history, you can go there. Warning: a lot of it is chemistry. You can skip that part. I did, and I have some knowledge of chemistry.

But I do want to focus on one nearly throwaway sentence from that article:

In 1946, the demand for nylon stockings could not be satisfied, which led to the Nylon riots.

The... Nylon... RIOTS?

This, I gotta learn more about.

The nylon riots were a series of disturbances at American stores created by a nylon stocking shortage.

Anyone else spot the inherent pun there? Anyone? I'll ruin it by explaining it: they had trouble stocking stockings.

During World War II, nylon was used extensively for parachutes and other war materials, such as airplane cords and ropes and the supply of nylon consumer goods was curtailed.

I'm just going to leave this here and hopefully remember to, at some point, research the effects of the rise of synthetic fabrics on the demonization of hemp. I can't help but think there was a connection, but it's irrelevant to this discussion.

The riots occurred between August 1945 and March 1946, when the War Production Board announced that the creation of Du Pont's nylon would shift its manufacturing from wartime material to nylon stockings, at the same time launching a promotional campaign.

On the other hand, I think I have a pretty good grasp on the link between overpromising stuff in promotional campaigns, and consumer riots.

It is imperative that I add: it sounds like there was a run on stockings.

In one of the worst disturbances, in Pittsburgh, 40,000 women queued up for 13,000 pairs of stockings, leading to fights breaking out.

If only we had camera phones then. Imagine the beautiful videos that would have resulted.

It took several months before Du Pont was able to ramp up production to meet demand, but until they did many women went without nylon stockings for months.

Imagine the sheer horror (pun intended).

During World War II, Japan stopped using supplies made out of silk, and so the United States had difficulty importing silk from Japan.

Pretty sure there were other reasons why importing silk from Japan wasn't possible during WWII.

Nylon stockings became increasingly popular on the black market, and sold for up to $20 per pair.

WTF, Wikipedia. Dollar amounts are useless unless we can compare them to today's dollars. Here, I'll help: $20 in 1945 is like $335 today.

There's more at the link. Maybe you'd heard of them; I had not, and despite the very real plight of the American stocking-wearers (or, technically, stocking-not-wearers) in the mid-1940s, I had a good laugh reading the section.

I have no idea what the current status of demand for nylon stockings is. I don't see them much, but I don't go to department stores or frequent the kinds of events where people would wear stockings or hose or pantyhose or whatever. Most of the ladies I see pictures of on the internet don't seem to wear them, but that may be selection bias, as most of the ladies I see pictures of on the internet don't wear much of anything.

No, I think people these days prefer "natural" fabrics, but not fur, though artificial fake fur can be made of nylon or nylon blends. I mostly see it in more bulk form, not fabric: fishing lines, zippers, even car parts. The first Wiki link above lists a bunch of uses. As a plastic, it's very versatile. But as a fabric? I can't remember the last time I could identify a pure nylon fabric. I think I had a raincoat made of it as a kid, but I'm not sure even of that.

I guess you could say that if you're looking for nylon stockings today, you might be hosed.

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